Editing Partial fraction decomposition (section)

===Powers in the denominator===
Using the preceding decomposition inductively one gets fractions of the form <math>\frac F {G^k},</math> with <math>\deg F < \deg G^k= k\deg G,</math> where {{mvar|G}} is an [[irreducible polynomial]]. If {{math|''k'' > 1}}, one can decompose further, by using that an irreducible polynomial is a [[square-free polynomial]], that is, <math>1</math> is a [[Polynomial greatest common divisor|greatest common divisor]] of the polynomial and its [[derivative]]. If <math>G'</math> is the derivative of {{mvar|G}}, [[Polynomial greatest common divisor#Bézout's identity and extended GCD algorithm|Bézout's identity]] provides polynomials {{mvar|C}} and {{mvar|D}} such that <math>CG + DG' = 1</math> and thus <math>F=FCG+FDG'.</math> Euclidean division of <math>FDG'</math> by <math>G</math>  gives polynomials <math>H_k</math> and <math>Q</math> such that <math>FDG' = QG + H_k</math> and <math>\deg H_k < \deg G.</math> Setting <math>F_{k-1}=FC+Q,</math> one gets
<math display="block">\frac F {G^k} = \frac{H_k}{G^k}+\frac{F_{k-1}}{G^{k-1}},</math>
with <math>\deg H_k <\deg G.</math>

Iterating this process with <math>\frac{F_{k-1}}{G^{k-1}}</math> in place of <math>\frac F{G^k}</math> leads eventually to the following theorem.